(Pre-)Algebras for Linguistics 5. Prelattices Carl Pollard - PowerPoint PPT Presentation

(Pre-)Algebras for Linguistics 5. Prelattices Carl Pollard Linguistics 680: Formal Foundations Autumn 2010 Carl Pollard (Pre-)Algebras for Linguistics

Prelattices A prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔� where � P, ⊑ , ⊓� is a lower semilattice and Carl Pollard (Pre-)Algebras for Linguistics

Prelattices A prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔� where � P, ⊑ , ⊓� is a lower semilattice and � P, ⊑ , ⊔� is an upper semilattice. Carl Pollard (Pre-)Algebras for Linguistics

Prelattices A prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔� where � P, ⊑ , ⊓� is a lower semilattice and � P, ⊑ , ⊔� is an upper semilattice. A bounded prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� where � P, ⊑ , ⊓ , ⊔� is a prelattice Carl Pollard (Pre-)Algebras for Linguistics

Prelattices A prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔� where � P, ⊑ , ⊓� is a lower semilattice and � P, ⊑ , ⊔� is an upper semilattice. A bounded prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� where � P, ⊑ , ⊓ , ⊔� is a prelattice ⊤ is a top Carl Pollard (Pre-)Algebras for Linguistics

Prelattices A prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔� where � P, ⊑ , ⊓� is a lower semilattice and � P, ⊑ , ⊔� is an upper semilattice. A bounded prelattice is a preordered algebra � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� where � P, ⊑ , ⊓ , ⊔� is a prelattice ⊤ is a top ⊥ is a bottom Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments associative u.t.e. Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments associative u.t.e. commutative u.t.e. Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e. Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e. ⊓ ( ⊔ ) is a glb (lub) operation Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e. ⊓ ( ⊔ ) is a glb (lub) operation Interdefinability: for all p, q ∈ P , p ⊓ q ≡ p iff p ⊑ q iff p ⊔ q ≡ q Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review) ⊓ and ⊔ are: monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e. ⊓ ( ⊔ ) is a glb (lub) operation Interdefinability: for all p, q ∈ P , p ⊓ q ≡ p iff p ⊑ q iff p ⊔ q ≡ q Absorption u.t.e.: ( p ⊔ q ) ⊓ q ≡ q ≡ ( p ⊓ q ) ⊔ q ; Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices Semidistributivity : For all a, b ∈ P : ( p ⊓ q ) ⊔ ( p ⊓ r ) ⊑ p ⊓ ( q ⊔ r ) Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices Semidistributivity : For all a, b ∈ P : ( p ⊓ q ) ⊔ ( p ⊓ r ) ⊑ p ⊓ ( q ⊔ r ) A prelattice is called distributive u.t.e if the inequality reverse to Semidistributivity holds: p ⊓ ( q ⊔ r ) ⊑ ( p ⊓ q ) ⊔ ( p ⊓ r ) so that in fact Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices Semidistributivity : For all a, b ∈ P : ( p ⊓ q ) ⊔ ( p ⊓ r ) ⊑ p ⊓ ( q ⊔ r ) A prelattice is called distributive u.t.e if the inequality reverse to Semidistributivity holds: p ⊓ ( q ⊔ r ) ⊑ ( p ⊓ q ) ⊔ ( p ⊓ r ) so that in fact p ⊓ ( q ⊔ r ) ≡ ( p ⊓ q ) ⊔ ( p ⊓ r ) Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices Semidistributivity : For all a, b ∈ P : ( p ⊓ q ) ⊔ ( p ⊓ r ) ⊑ p ⊓ ( q ⊔ r ) A prelattice is called distributive u.t.e if the inequality reverse to Semidistributivity holds: p ⊓ ( q ⊔ r ) ⊑ ( p ⊓ q ) ⊔ ( p ⊓ r ) so that in fact p ⊓ ( q ⊔ r ) ≡ ( p ⊓ q ) ⊔ ( p ⊓ r ) Theorem : a prelattice is distributive u.t.e. iff the following equivalence holds for all a, b, c ∈ P (obtained from the one above by interchanging ⊓ and ⊔ ): p ⊔ ( q ⊓ r ) ≡ ( p ⊔ q ) ⊓ (p ⊔ r ) Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations Suppose � P, ⊑ , ⊓ , → , ⊥� is a heyting presemilattice with a bottom element ⊥ . Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P , ¬ p ≡ p → ⊥ Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations Suppose � P, ⊑ , ⊓ , → , ⊥� is a heyting presemilattice with a bottom element ⊥ . Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P , ¬ p ≡ p → ⊥ Clearly a pseudocomplement operation is antitonic. It’s easy to show that ¬⊥ is a top. Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations Suppose � P, ⊑ , ⊓ , → , ⊥� is a heyting presemilattice with a bottom element ⊥ . Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P , ¬ p ≡ p → ⊥ Clearly a pseudocomplement operation is antitonic. It’s easy to show that ¬⊥ is a top. It’s easy to show that for all p ∈ P , p ⊑ ¬ ( ¬ p ). Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations Suppose � P, ⊑ , ⊓ , → , ⊥� is a heyting presemilattice with a bottom element ⊥ . Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P , ¬ p ≡ p → ⊥ Clearly a pseudocomplement operation is antitonic. It’s easy to show that ¬⊥ is a top. It’s easy to show that for all p ∈ P , p ⊑ ¬ ( ¬ p ). A pseudocomplement operation is called a complement operation provided, for all a ∈ P , ¬ ( ¬ a ) ⊑ a , so that in fact ¬ ( ¬ a ) ≡ a . Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras A pre-heyting algebra is a preordered algebra � P, ⊑ , ⊓ , ⊔ , → , ¬ , ⊤ , ⊥� where: � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� is a bounded prelattice Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras A pre-heyting algebra is a preordered algebra � P, ⊑ , ⊓ , ⊔ , → , ¬ , ⊤ , ⊥� where: � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� is a bounded prelattice � P, ⊑ , ⊓ , →� is a heyting presemilattice Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras A pre-heyting algebra is a preordered algebra � P, ⊑ , ⊓ , ⊔ , → , ¬ , ⊤ , ⊥� where: � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� is a bounded prelattice � P, ⊑ , ⊓ , →� is a heyting presemilattice ¬ is a pseudocomplement operation. Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras A pre-heyting algebra is a preordered algebra � P, ⊑ , ⊓ , ⊔ , → , ¬ , ⊤ , ⊥� where: � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� is a bounded prelattice � P, ⊑ , ⊓ , →� is a heyting presemilattice ¬ is a pseudocomplement operation. Theorem: pre-heyting algebras are distributive u.t.e. Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras A pre-heyting algebra is a preordered algebra � P, ⊑ , ⊓ , ⊔ , → , ¬ , ⊤ , ⊥� where: � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� is a bounded prelattice � P, ⊑ , ⊓ , →� is a heyting presemilattice ¬ is a pseudocomplement operation. Theorem: pre-heyting algebras are distributive u.t.e. Pre-heyting algebras are algebraic models of a kind of logic called intuitionistic propositional logic . Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras A pre-heyting algebra is a preordered algebra � P, ⊑ , ⊓ , ⊔ , → , ¬ , ⊤ , ⊥� where: � P, ⊑ , ⊓ , ⊔ , ⊤ , ⊥� is a bounded prelattice � P, ⊑ , ⊓ , →� is a heyting presemilattice ¬ is a pseudocomplement operation. Theorem: pre-heyting algebras are distributive u.t.e. Pre-heyting algebras are algebraic models of a kind of logic called intuitionistic propositional logic . A pre-boolean algebra is a pre-heyting algebra satisfying either of the following (equivalent!) conditions: The pseudocomplement operation ¬ is a complement operation, i.e. for all p ∈ P , ¬ ( ¬ p ) ⊑ p . Carl Pollard (Pre-)Algebras for Linguistics